As a young mathematician, one learns that the derivative and integral are kind of like inverse operations, but not quite. The two fundamental theorems of calculus require some additional assumptions to go through. Moreover, the maps $f \mapsto f'$ and $f \mapsto F$ (where $F(x) = \int_a^x f(t) dt$ for some fixed $a$) both fail to be injective and surjective on many function spaces. Yet, it remains a useful intuition to think of integration and differentiation as opposite, if not inverse, operations.
Are there other examples of this near inverse behavior in mathematics? Is there a more general phenomenon like this that integration and differentiation are just examples of?